MinT - Best Known (143, 254, s)-Nets in Base 4

Best Known (143, 254, s)-Nets in Base 4

(143, 254, 137)-Net over F₄ — Constructive and digital

Digital (143, 254, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 70, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 184, 104)-net over F₄, using
  - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
      - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(143, 254, 272)-Net over F₄ — Digital

Digital (143, 254, 272)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(143, 254, 4138)-Net in Base 4 — Upper bound on s

There is no (143, 254, 4139)-net in base 4, because

1 times m-reduction [i] would yield (143, 253, 4139)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 212 015333 813275 464067 978285 883550 957758 937202 807930 316513 832370 195588 962498 346865 915710 945274 796821 759594 046928 901464 660258 682228 803428 415991 091834 880000 > 4²⁵³ [i]